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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(151,925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("925.151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 151.12
Character \(\chi\) \(=\) 925.151
Dual form 925.2.bb.b.876.12

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50953 - 1.79899i) q^{2} +(-0.670695 + 0.562780i) q^{3} +(-0.610385 - 3.46166i) q^{4} +2.05611i q^{6} +(4.17567 + 1.51982i) q^{7} +(-3.08132 - 1.77900i) q^{8} +(-0.387834 + 2.19951i) q^{9} +(-2.58382 + 4.47531i) q^{11} +(2.35754 + 1.97821i) q^{12} +(4.66864 - 0.823207i) q^{13} +(9.03746 - 5.21778i) q^{14} +(-1.24561 + 0.453366i) q^{16} +(1.88224 + 0.331889i) q^{17} +(3.37146 + 4.01795i) q^{18} +(1.95658 + 2.33176i) q^{19} +(-3.65593 + 1.33065i) q^{21} +(4.15068 + 11.4039i) q^{22} +(-8.02299 + 4.63208i) q^{23} +(3.06782 - 0.540939i) q^{24} +(5.56653 - 9.64151i) q^{26} +(-2.29102 - 3.96817i) q^{27} +(2.71234 - 15.3824i) q^{28} +(-2.53625 - 1.46431i) q^{29} -5.61699i q^{31} +(1.36913 - 3.76165i) q^{32} +(-0.785658 - 4.45569i) q^{33} +(3.43836 - 2.88513i) q^{34} +7.85071 q^{36} +(4.05888 - 4.53051i) q^{37} +7.14833 q^{38} +(-2.66795 + 3.17954i) q^{39} +(0.132785 + 0.753063i) q^{41} +(-3.12492 + 8.58565i) q^{42} -0.369905i q^{43} +(17.0691 + 6.21265i) q^{44} +(-3.77791 + 21.4256i) q^{46} +(-5.66373 - 9.80987i) q^{47} +(0.580281 - 1.00508i) q^{48} +(9.76406 + 8.19302i) q^{49} +(-1.44919 + 0.836689i) q^{51} +(-5.69933 - 15.6588i) q^{52} +(4.35631 - 1.58557i) q^{53} +(-10.5971 - 1.86855i) q^{54} +(-10.1628 - 12.1116i) q^{56} +(-2.62453 - 0.462776i) q^{57} +(-6.46283 + 2.35228i) q^{58} +(-1.52849 - 4.19950i) q^{59} +(-2.71555 + 0.478824i) q^{61} +(-10.1049 - 8.47904i) q^{62} +(-4.96233 + 8.59501i) q^{63} +(-6.02598 - 10.4373i) q^{64} +(-9.20173 - 5.31262i) q^{66} +(4.24119 + 1.54367i) q^{67} -6.71825i q^{68} +(2.77414 - 7.62190i) q^{69} +(-3.35412 + 2.81444i) q^{71} +(5.10798 - 6.08746i) q^{72} +2.68113 q^{73} +(-2.02333 - 14.1408i) q^{74} +(6.87750 - 8.19628i) q^{76} +(-17.5908 + 14.7605i) q^{77} +(1.69261 + 9.59925i) q^{78} +(0.687516 - 1.88893i) q^{79} +(-2.52648 - 0.919562i) q^{81} +(1.55520 + 0.897894i) q^{82} +(-0.321200 + 1.82162i) q^{83} +(6.83778 + 11.8434i) q^{84} +(-0.665457 - 0.558384i) q^{86} +(2.52514 - 0.445249i) q^{87} +(15.9232 - 9.19324i) q^{88} +(2.11794 + 5.81901i) q^{89} +(20.7458 + 3.65805i) q^{91} +(20.9318 + 24.9456i) q^{92} +(3.16113 + 3.76729i) q^{93} +(-26.1975 - 4.61932i) q^{94} +(1.19871 + 3.29344i) q^{96} +(-2.46710 + 1.42438i) q^{97} +(29.4784 - 5.19783i) q^{98} +(-8.84141 - 7.41882i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.50953 − 1.79899i 1.06740 − 1.27208i 0.106761 − 0.994285i \(-0.465952\pi\)
0.960640 − 0.277795i \(-0.0896036\pi\)
\(3\) −0.670695 + 0.562780i −0.387226 + 0.324921i −0.815531 − 0.578713i \(-0.803555\pi\)
0.428305 + 0.903634i \(0.359111\pi\)
\(4\) −0.610385 − 3.46166i −0.305192 − 1.73083i
\(5\) 0 0
\(6\) 2.05611i 0.839404i
\(7\) 4.17567 + 1.51982i 1.57826 + 0.574438i 0.974824 − 0.222977i \(-0.0715775\pi\)
0.603431 + 0.797415i \(0.293800\pi\)
\(8\) −3.08132 − 1.77900i −1.08941 − 0.628973i
\(9\) −0.387834 + 2.19951i −0.129278 + 0.733172i
\(10\) 0 0
\(11\) −2.58382 + 4.47531i −0.779051 + 1.34936i 0.153439 + 0.988158i \(0.450965\pi\)
−0.932489 + 0.361197i \(0.882368\pi\)
\(12\) 2.35754 + 1.97821i 0.680563 + 0.571060i
\(13\) 4.66864 − 0.823207i 1.29485 − 0.228317i 0.516575 − 0.856242i \(-0.327207\pi\)
0.778273 + 0.627926i \(0.216096\pi\)
\(14\) 9.03746 − 5.21778i 2.41536 − 1.39451i
\(15\) 0 0
\(16\) −1.24561 + 0.453366i −0.311403 + 0.113341i
\(17\) 1.88224 + 0.331889i 0.456509 + 0.0804949i 0.397174 − 0.917743i \(-0.369991\pi\)
0.0593351 + 0.998238i \(0.481102\pi\)
\(18\) 3.37146 + 4.01795i 0.794661 + 0.947040i
\(19\) 1.95658 + 2.33176i 0.448870 + 0.534942i 0.942267 − 0.334862i \(-0.108690\pi\)
−0.493398 + 0.869804i \(0.664245\pi\)
\(20\) 0 0
\(21\) −3.65593 + 1.33065i −0.797789 + 0.290371i
\(22\) 4.15068 + 11.4039i 0.884928 + 2.43132i
\(23\) −8.02299 + 4.63208i −1.67291 + 0.965855i −0.706913 + 0.707300i \(0.749913\pi\)
−0.965997 + 0.258555i \(0.916754\pi\)
\(24\) 3.06782 − 0.540939i 0.626216 − 0.110419i
\(25\) 0 0
\(26\) 5.56653 − 9.64151i 1.09169 − 1.89086i
\(27\) −2.29102 − 3.96817i −0.440907 − 0.763674i
\(28\) 2.71234 − 15.3824i 0.512584 − 2.90701i
\(29\) −2.53625 − 1.46431i −0.470970 − 0.271915i 0.245676 − 0.969352i \(-0.420990\pi\)
−0.716646 + 0.697437i \(0.754324\pi\)
\(30\) 0 0
\(31\) − 5.61699i − 1.00884i −0.863458 − 0.504421i \(-0.831706\pi\)
0.863458 − 0.504421i \(-0.168294\pi\)
\(32\) 1.36913 − 3.76165i 0.242030 − 0.664971i
\(33\) −0.785658 − 4.45569i −0.136766 − 0.775636i
\(34\) 3.43836 − 2.88513i 0.589675 − 0.494796i
\(35\) 0 0
\(36\) 7.85071 1.30845
\(37\) 4.05888 − 4.53051i 0.667276 − 0.744811i
\(38\) 7.14833 1.15961
\(39\) −2.66795 + 3.17954i −0.427214 + 0.509134i
\(40\) 0 0
\(41\) 0.132785 + 0.753063i 0.0207376 + 0.117609i 0.993419 − 0.114534i \(-0.0365375\pi\)
−0.972682 + 0.232143i \(0.925426\pi\)
\(42\) −3.12492 + 8.58565i −0.482186 + 1.32479i
\(43\) − 0.369905i − 0.0564100i −0.999602 − 0.0282050i \(-0.991021\pi\)
0.999602 − 0.0282050i \(-0.00897912\pi\)
\(44\) 17.0691 + 6.21265i 2.57327 + 0.936593i
\(45\) 0 0
\(46\) −3.77791 + 21.4256i −0.557022 + 3.15903i
\(47\) −5.66373 − 9.80987i −0.826140 − 1.43092i −0.901045 − 0.433726i \(-0.857199\pi\)
0.0749052 − 0.997191i \(-0.476135\pi\)
\(48\) 0.580281 − 1.00508i 0.0837564 − 0.145070i
\(49\) 9.76406 + 8.19302i 1.39487 + 1.17043i
\(50\) 0 0
\(51\) −1.44919 + 0.836689i −0.202927 + 0.117160i
\(52\) −5.69933 − 15.6588i −0.790355 − 2.17148i
\(53\) 4.35631 − 1.58557i 0.598385 − 0.217794i −0.0250282 − 0.999687i \(-0.507968\pi\)
0.623414 + 0.781892i \(0.285745\pi\)
\(54\) −10.5971 − 1.86855i −1.44208 − 0.254277i
\(55\) 0 0
\(56\) −10.1628 − 12.1116i −1.35807 − 1.61848i
\(57\) −2.62453 − 0.462776i −0.347628 − 0.0612962i
\(58\) −6.46283 + 2.35228i −0.848612 + 0.308869i
\(59\) −1.52849 − 4.19950i −0.198993 − 0.546728i 0.799556 − 0.600592i \(-0.205068\pi\)
−0.998548 + 0.0538642i \(0.982846\pi\)
\(60\) 0 0
\(61\) −2.71555 + 0.478824i −0.347690 + 0.0613071i −0.344766 − 0.938689i \(-0.612042\pi\)
−0.00292379 + 0.999996i \(0.500931\pi\)
\(62\) −10.1049 − 8.47904i −1.28333 − 1.07684i
\(63\) −4.96233 + 8.59501i −0.625195 + 1.08287i
\(64\) −6.02598 − 10.4373i −0.753248 − 1.30466i
\(65\) 0 0
\(66\) −9.20173 − 5.31262i −1.13265 − 0.653938i
\(67\) 4.24119 + 1.54367i 0.518144 + 0.188589i 0.587837 − 0.808979i \(-0.299980\pi\)
−0.0696928 + 0.997569i \(0.522202\pi\)
\(68\) − 6.71825i − 0.814707i
\(69\) 2.77414 − 7.62190i 0.333968 − 0.917568i
\(70\) 0 0
\(71\) −3.35412 + 2.81444i −0.398061 + 0.334013i −0.819744 − 0.572731i \(-0.805884\pi\)
0.421683 + 0.906744i \(0.361440\pi\)
\(72\) 5.10798 − 6.08746i 0.601982 − 0.717414i
\(73\) 2.68113 0.313803 0.156902 − 0.987614i \(-0.449849\pi\)
0.156902 + 0.987614i \(0.449849\pi\)
\(74\) −2.02333 − 14.1408i −0.235207 − 1.64384i
\(75\) 0 0
\(76\) 6.87750 − 8.19628i 0.788903 − 0.940178i
\(77\) −17.5908 + 14.7605i −2.00466 + 1.68211i
\(78\) 1.69261 + 9.59925i 0.191650 + 1.08690i
\(79\) 0.687516 − 1.88893i 0.0773516 − 0.212522i −0.894989 − 0.446087i \(-0.852817\pi\)
0.972341 + 0.233566i \(0.0750393\pi\)
\(80\) 0 0
\(81\) −2.52648 − 0.919562i −0.280720 − 0.102174i
\(82\) 1.55520 + 0.897894i 0.171743 + 0.0991558i
\(83\) −0.321200 + 1.82162i −0.0352563 + 0.199949i −0.997348 − 0.0727768i \(-0.976814\pi\)
0.962092 + 0.272725i \(0.0879250\pi\)
\(84\) 6.83778 + 11.8434i 0.746063 + 1.29222i
\(85\) 0 0
\(86\) −0.665457 − 0.558384i −0.0717580 − 0.0602121i
\(87\) 2.52514 − 0.445249i 0.270723 − 0.0477358i
\(88\) 15.9232 − 9.19324i 1.69741 − 0.980003i
\(89\) 2.11794 + 5.81901i 0.224502 + 0.616813i 0.999892 − 0.0146684i \(-0.00466927\pi\)
−0.775391 + 0.631482i \(0.782447\pi\)
\(90\) 0 0
\(91\) 20.7458 + 3.65805i 2.17475 + 0.383468i
\(92\) 20.9318 + 24.9456i 2.18229 + 2.60075i
\(93\) 3.16113 + 3.76729i 0.327794 + 0.390650i
\(94\) −26.1975 − 4.61932i −2.70206 − 0.476446i
\(95\) 0 0
\(96\) 1.19871 + 3.29344i 0.122343 + 0.336135i
\(97\) −2.46710 + 1.42438i −0.250496 + 0.144624i −0.619991 − 0.784609i \(-0.712864\pi\)
0.369496 + 0.929233i \(0.379531\pi\)
\(98\) 29.4784 − 5.19783i 2.97776 − 0.525060i
\(99\) −8.84141 − 7.41882i −0.888595 − 0.745620i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 925.2.bb.b.151.12 72
5.2 odd 4 925.2.ba.c.299.12 72
5.3 odd 4 925.2.ba.b.299.1 72
5.4 even 2 185.2.w.a.151.1 yes 72
37.25 even 18 inner 925.2.bb.b.876.12 72
185.62 odd 36 925.2.ba.b.99.1 72
185.69 odd 36 6845.2.a.u.1.2 36
185.79 odd 36 6845.2.a.t.1.35 36
185.99 even 18 185.2.w.a.136.1 ✓ 72
185.173 odd 36 925.2.ba.c.99.12 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.1 ✓ 72 185.99 even 18
185.2.w.a.151.1 yes 72 5.4 even 2
925.2.ba.b.99.1 72 185.62 odd 36
925.2.ba.b.299.1 72 5.3 odd 4
925.2.ba.c.99.12 72 185.173 odd 36
925.2.ba.c.299.12 72 5.2 odd 4
925.2.bb.b.151.12 72 1.1 even 1 trivial
925.2.bb.b.876.12 72 37.25 even 18 inner
6845.2.a.t.1.35 36 185.79 odd 36
6845.2.a.u.1.2 36 185.69 odd 36